A collective-rotation noise can be described as

Here θ is the noise parameter, which fluctuates with time. That is, if a | 0〉 state is delivered through channel with collective-rotation noise, the state will change into cosθ | 0〉 + sin θ | 1〉 ; while, a | 1〉 state will change into − sinθ | 0〉 + cosθ | 1〉 . For collective-rotation noise, | ϕ ⁺ 〉 and | ψ ⁻ 〉 are invariant. Therefore, logical qubits immune to collective-rotation noise are described respectively as

Now let us briefly describe the DSQC protocol based on hyperentanglement against collective-rotation noise. We suppose that the sender, Bob, wants to transmit his secret message to the legitimate receiver, say Alice. Alice and Bob have the secret w-bit string identity (ID) representing their identities. Suppose that Bob’ s secret message is a series of classical 0 or 1 numbers in order, called M.

For a hyperentangled Bell state

if particle B is delivered through quantum channel with collective- rotation noise, to resist the effect of noise, the sender should represent particle B as a logical quantum state, that is | ϕ ⁺ , | 0〉 _B should be expressed as | ϕ ⁺ 〉 _B1B2, and | 1〉 _B is expressed as | ψ ⁻ 〉 _B1B2. Therefore, the logical hyperentangled Bell state should be expressed as

Step 1 Alice prepares N ordered states. We denote the ordered states as {[P₁(A), P₁(B₁), P₁(B₂)], [P₂(A), P₂(B₁), P₂(B₂)], … , [P_N(A), P_N(B₁), P_N(B₂)]}, where the subscript indicates the order of each state in the sequence, and A, B₁, B₂ represent the three particles of each state. Alice takes particle A from each state to form an ordered particle sequence {P₁(A), P₂(A), … , P_N(A)}, called the S_A sequence. The remaining particles constitute an S_B sequence, denoted as {P₁(B₁), P₁(B₂), P₂(B₁), P₂(B₂), … , P_N(B₁), P_N(B₂)}. The idea of block transmission is similar to the original protocol.^[7] Alice prepares another 2N/3 ordered states, which form a sequence S_T. The S_T is denoted as {T₁(A), T₁(B₁), T₁(B₂), T₂(A), T₂(B₁), T₂(B₂), … , T_N(A), T_N(B₁), T_N(B₂)}, which is used as decoy photons.

Step 2 Alice inserts the decoy photons S_T into sequence S_B according to ID, forming a new sequence . For example, if the i-th bit of ID is 0, Alice inserts the i-th photon of S_T behind the i-th photon of S_B; if the i-th bit of ID is 1, Alice inserts the i-th photon of S_T before the i-th photon of S_B. Only Alice and Bob know the positions of these decoy photons. Then Alice sends to Bob.

Step 3 After Bob receives , Bob extracts the decoy photons from according to ID. Bob performs three single-qubit measurements on the S_T extracted respectively. In the ideal cases, each result should be in one of the four | 000〉 , | 011〉 , | 101〉 , | 110〉 states with equal probability. If the error rate is low enough Bob believes that Alice is legal and no eavesdropping exists. In this condition, the communication goes on. Otherwise Bob interrupts it.

Step 4 Alice and Bob perform S ⊗ S ⊗ S and H ⊗ H ⊗ H operations on particles AB₁B₂ in turn respectively. Then the combined system of AB₁B₂ will change into

Here, the phase gate S and Hadamard gate H are represented respectively as

Step 5 Bob publishes the photons that he does not receive in S_B, Alice discards the corresponding photons in S_A. Therefore, photons in S_B will be always one-to-one matching with photons in S_A.

Step 6 Alice measures photons in S_A with basis Z^S = {| a₁〉 , | a₂〉 } (measurement basis in spatial-mode degree of freedom). Then according to ID, Alice measures photons in S_A with basis Z^P = {| 0〉 , | 1〉 } or X^P = {| + 〉 _P, | − 〉 _P} (non-orthodox measurement basis in polarization degree of freedom) in order. That is, if the bit of ID is “ 0” , Alice measures the corresponding photon of S_A with the Z^P basis; otherwise, Alice uses the X^P basis. If the measurement result is | 0〉 or | + 〉 , Alice refers to the result as “ 0” ; otherwise, Alice refers to the result as “ 1” . By measuring photons in S_A with the basis in spatial-mode and polarization degree of freedom, Alice will obtain the measurement result | a₁0〉 , | a₂0〉 , | a₁1〉 , | a₂1〉 or | a₁+ 〉 , | a₂+ 〉 , | a₁− 〉 , | a₂− 〉 with equal probability. At the same time the states of particles B₁B₂ are fixed: they are | b₁c₁〉 | 01〉 , | b₂c₂〉 | 01〉 , | b₁c₁〉 | 10〉 , | b₂c₂〉 | 10〉 or | b₁c₁〉 | + − 〉 (| − + 〉 ), | b₂c₂〉 | + − 〉 (| − + 〉 ), | b₁c₁〉 | + + 〉 (| − − 〉 ), | b₂c₂〉 | + + 〉 (| − − 〉 ) respectively. And if the measurement result of Alice is | a₁0〉 , | a₂0〉 , | a₁1〉 , | a₂1〉 , the measurement result of Bob must be | b₁c₁〉 | 01〉 , | b₂c₂〉 | 01〉 , | b₁c₁〉 | 10〉 , | b₂c₂〉 | 10〉 respectively; and if the measurement result of Alice is | a₁+ 〉 , | a₂+ 〉 , | a₁− 〉 , | a₂− 〉 , the measurement result of Bob must be | b₁c₁〉 | + − 〉 (| − + 〉 ), | b₂c₂〉 | + − 〉 (| − + 〉 ), | b₁c₁〉 | + + 〉 (| − − 〉 ), | b₂c₂〉 | + + 〉 (| − − 〉 ) respectively. If we use binary string {00, 01, 10, 11} to express {| a₁0〉 , | a₂0〉 , | a₁1〉 , | a₂1〉 }, {| a₁+ 〉 , | a₂+ 〉 , | a₁− 〉 , | a₂− 〉 }, {| b₁c₁〉 | 01〉 , | b₂c₂〉 | 01〉 , | b₁c₁〉 | 10〉 , | b₂c₂〉 | 10〉 } or {| b₁c₁〉 | + − 〉 (| − + 〉 ), | b₂c₂〉 | + − 〉 (| − + 〉 ), | b₁c₁〉 | + + 〉 (| − − 〉 ), | b₂c₂〉 | + + 〉 (| − − 〉 )}, Alice and Bob will obtain an identical binary string C, which is a real random number.

Step 7 Bob also measures photons B₁B₂ with the same method as Alice. If the measurement result is | 01〉 , | + − 〉 or | − + 〉 , Bob denotes it as “ 0” ; otherwise as “ 1” . By doing so, Bob will obtain C.

Step 8 Assuming the length of C to be L, Bob divides the secret message M into several parts. Each part is termed M_i (i = 1, 2, … ), which includes the classical secret information about L bits. Bob encrypts the secret message M₁ with C bit by bit using XOR operation and obtains C₁. That is, C₁ = M₁ XOR C. Bob publishes C₁.

Step 9 Alice decrypts C₁ with C bit by bit using XOR operation and obtains M₁. That is, M₁ = C₁ XOR C.

Step 10 Alice stores secret message M₁.

Step 11 Alice and Bob begin the next transmission, until all secret messages are finished.

Because the DSQC scheme is designed to be immune to collective-rotation noise, the eavesdropping will be found more easily. Thus, the security of the protocol is mainly guaranteed by the identity string ID (shared by Alice and Bob) and the eavesdropping detection strategy based on in polarization degree of freedom). Because identity authentication is implemented based on the identity string, the protocol will not be threatened by a man-in-the-middle attack. Next, we analyze the eavesdropping detection rate of the protocol.

3.2.1. Analysis of the detection rate

In the DSQC scheme immune to collective-rotation noise, to reduce the state one needs to prepare the

state that is used as decoy photons to detect eavesdropping (that is, only one state is needed). Because the positions of decoy photons are secret, Eve cannot discriminate between decoy photons and information photons (photons in S_B).

According to Stingspring dilation theorem, Eve’ s eavesdropping can be implemented by a unitary operation E acting on a bigger Hilbert space H_AB1B2 ⊗ H_E. The eavesdropping can be represented as

Here, | x〉 _E is the initial state of Eve’ s auxiliary particle and | a| ² + | b| ² = 1, | m| ² + | n| ² = 1. | x₀ 〉 _E, | x₁〉 _E, | y₀〉 _E, | y₁〉 _E are pure states uniquely determined by unitary operation E.

The unitary operation E of Eve can be represented as

The complex numbers a, b, m, n satisfy EE^† = I. Therefore, | a| ² = | m| ² and | b| ² = | n| ² can be obtained.

After Eve’ s eavesdropping, the combined system changes into

After Bob receives (mixture of S_B and S_T), Bob extracts S_T and performs three single-qubit measurements on S_T. The probability without an eavesdropper is

Because | a| ² = | m| ² and | b| ² = | n| ², then

Let | a| ² = s and | b| ² = t, then t + s = 1, we can obtain

Therefore, the error rate of each qubit is

which can also be seen as the lower bound of the detection rate of each qubit eavesdropped:

According to the theory of von Neumann entropy, the maximum amount of information contained in qubit | 0〉 is termed

And the maximum amount of information contained in qubit | 1〉 is termed

For a qubit transmitting in a quantum channel, it will be in | 0〉 or | 1〉 state with equal probability (p = 0.5); therefore, the total information that Eve can eavesdrop in a qubit will be

According to Eq.  (18), we obtain the relationship between s and d as follows:

Let

and substitute s into Eq.  (22), then we will obtain that

and

Therefore,

We obtain the relationship between I (the maximum amount of information) and d (the detection rate) as follows:

From Fig.  2 we find that when d = 0.5, then I = 1, which is the maximum amount of information Eve can eavesdrop. That is, if Eve performs an attack by using ancillary particles, when she obtains the maximum amount of information on one qubit, the detection rate or qubit error rate reaches 50%.

Fig.  2.

	Figure Option ViewDownloadNew Window
	Fig.  2. Relationship between I and d.

The original decoy photon idea is proposed by Li et al. in Refs.  [63] and [64]. In Refs.  [63] and [64], a group of non-orthodox photons are used as decoy photons. In our protocol, the unique qubits

are used as decoy photons. However, the unique qubits will not reduce the security of this detection strategy, the reasons are shown as follows: i) because the position of S_T in is secret, Eve cannot obtain the secret by any measurement; ii) because Alice and Bob measure the information photons (S_B) with Z^P or X^P basis according to ID (which is shared by Alice and Bob only), the decoy photons will be measured with three single-qubit measurements , to eavesdrop on the secret, Eve may run a great risk of measuring decoy photons with the X^P basis. That is, to eavesdrop on the secret, Eve has to measure photons with the X^P basis sometimes, however, because Eve does not know which photons should be measured with the X^P basis and which photons should be measured with the Z^P basis, she may measure decoy photons with the X^P basis by mistake, which will be found by Alice and Bob easily.

A collective-dephasing noise can be described as

Here ϕ is the noise parameter, which fluctuates with time. That is, if a | 0〉 state is delivered through the channel with collective-dephasing noise, the state will not change; while a | 1〉 state will change into e^iϕ | 1〉 . Because two physical qubits with anti-parallel parity obtain the same phase factor iϕ , to defend against collective-dephasing noise, two anti-parallel qubits can form a logical qubit. The logical qubits immune to collective-dephasing noise can be described by Eq.  (8), and given as

Now we briefly describe the DSQC protocol based on hyperentanglement against a collective-dephasing noise. Suppose that Bob is the sender and Alice is the legitimate receiver. Alice and Bob have the secret w-bit string ID representing their identities. Bob’ s secret message is a series of classical 0 or 1 numbers in order, called M.

For a hyperentangled Bell state

if particle B is delivered through the quantum channel with collective-dephasing noise, to resist the effect of noise, the sender should represent particle B as a logical quantum state, that is, | 0〉 _B should be expressed as | 0〉 _B1 | 1〉 _B2; | 1〉 _B is expressed as | 1〉 _B1 | 0〉 _B2. Therefore, the logical hyperentangled Bell state should be expressed as

Step 1 Alice prepares N ordered states. We denote the ordered states as {[P₁(A), P₁(B₁), P₁(B₂)], [P₂(A), P₂(B₁), P₂(B₂)], … , [P_N(A), P_N(B₁), P_N(B₂)]}. Alice takes particle A from each state to form an ordered particle sequence S_A sequence. The remaining particles constitute an S_B sequence. It is similar to the original protocol.^[7]

Step 2 According to the ID, Alice prepares a qubit sequence S_T. As did in Refs.  [63] and [64], Alice uses S_T as decoy photons. The rule is that she randomly prepares the i-th qubit of S_T in | 0〉 or | 1〉 state if the i-th bit of the ID is 0; or else, she randomly prepares the qubit in | + 〉 or | − 〉 state. Alice inserts the decoy sequence S_T into the sequence S_B according to the ID, forming a new sequence , which is similar to Step 2 in the DSQC scheme against collective-rotation noise. Only Alice and Bob know the positions of these decoy photons. Then Alice sends to Bob.

Step 3 After Bob receives , Bob extracts decoy photons from and measures it with Z^P = {| 0〉 , | 1〉 } or X_P = {| + 〉 _P, | − 〉 _P} basis according to the ID. If the error rate is low enough Bob believes that Alice is legal and no eavesdropping exists. In this condition, the communication goes on. Otherwise Bob interrupts it.

Step 4 Bob publishes the photons that he does not receive in S_B, Alice discards the corresponding photons in S_A.

Step 5 Alice first measures photons in S_A with basis Z^S = {| a₁〉 , | a₂〉 }. Then according to the ID, Alice measures photons in S_A with basis Z^P = {| 0〉 , | 1〉 } or X^P = {| + 〉 _P, | − 〉 _P}. Like Step 6 in the DSQC scheme against collective-rotation noise, Alice and Bob will obtain an identical binary string C, which is a real random number.

Step 6 The next steps are the same as Steps 7– 11 in the DSQC scheme against collective-rotation noise, which will not be described further here.

Because the DSQC scheme is designed to be immune to collective-dephasing noise, the eavesdropping will be found more easily. Thus, the security of the protocol is mainly guaranteed by the identity string ID (shared by Alice and Bob) and the eavesdropping detection strategy based on | 0〉 , | 1〉 , | + 〉 , and | − 〉 states. Because identity authentication is implemented based on the identity string, the protocol will not be threatened by a man-in-the-middle attack. Next, we analyze the eavesdropping detection rate of the protocol.